三维单向拉伸马蹄的寻找及其在Compass行走模型中的应用

计算物理 ›› 2016, Vol. 33 ›› Issue (1): 108-116.

三维单向拉伸马蹄的寻找及其在Compass行走模型中的应用

李清都^1,2, 赵武斌², 杨芳艳²

1. 重庆邮电大学工业物联网与网络化控制教育部重点实验室, 重庆 400065;
2. 重庆邮电大学复杂系统分析与控制研究中心, 重庆 400065

收稿日期:2014-11-21 修回日期:2015-04-07 出版日期:2016-01-25 发布日期:2016-01-25
作者简介:李清都(198-),男,湖北襄樊,博士,教授,主要从事混沌、电路与系统、非线性动力系统等研究,E-mail:liqd@cqupt.edu.cn
基金资助:
国家自然科学基金（61104150）、重庆市杰出青年科学基金（cstc2013jcyjjq40001）及重庆市教委科学技术研究项目（KJ130517）资助

3D Topological Horseshoes with 1-Directional Expansion and Application in Compass Walking Model

LI Qingdu^1,2, ZHAO Wubin², YANG Fangyan²

1. Key Laboratory of Industrial Internet of Things & Networked Control of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China;
2. Research Center of Analysis and Control for Complex Systems, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received:2014-11-21 Revised:2015-04-07 Online:2016-01-25 Published:2016-01-25

摘要/Abstract

摘要： 对三维映射，因维数较高，混沌吸引子结构复杂.根据混沌吸引子维数相对较低的特点，在选定不稳定周期轨的附近，对吸引子的局部进行曲面拟合，提出了三维空间中单向拉伸拓扑马蹄的“降维-升维”寻找方法，并实现为MATLAB工具箱.以Compass行走模型为例，详细介绍拓扑马蹄的寻找过程，判定了混沌步态的存在性，刻画了混沌不变集的部分结构.

关键词: 拓扑马蹄, 混沌, 被动行走, Compass 行走模型

Abstract: It is hard to apply topological horseshoe in 3D maps due to high dimension and complex structure since dimension of a chaotic attractor is often lower than its state space. We fit a surface with attractor near a selected unstable periodic orbit, and present a "dimension reduction" method to find topological horseshoes with one-directional expansion in 3D space. It is realized with a MATLAB toolbox. Compass walking model is shown to verify effectiveness, and illustrates detailed procedure for finding a topological horseshoe. It successfully verifys existence of chaotic gait and analyzes chaotic invariant set with horseshoe.

Key words: topological horseshoe, chaos, passive-walking, Compass walking model

中图分类号:

李清都, 赵武斌, 杨芳艳. 三维单向拉伸马蹄的寻找及其在Compass行走模型中的应用[J]. 计算物理, 2016, 33(1): 108-116.

LI Qingdu, ZHAO Wubin, YANG Fangyan. 3D Topological Horseshoes with 1-Directional Expansion and Application in Compass Walking Model[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2016, 33(1): 108-116.

参考文献

[1] MISCHAIKOW K, MROZEK M. Chaos in the Lorenz equations: a computer-assisted proof[J]. Bulletin of the American Mathematical Society, 1995, 32(1): 66-72.
[2] LI Q, YANG X S, CHEN S. Hyperchaos in a spacecraft power system[J]. International Journal of Bifurcation and Chaos, 2011,21: 1719-1726.
[3] YANG X S. Horseshoe chaos in a simple memristive circuit[J]. International Journal of Bifurcation and Chaos, 2009,19: 1127-1145.
[4] LI Q, YANG X S. Topological horseshoe in a chaotic system with no equilibria[J]. International Journal of Bifurcation and Chaos, 2010, 20: 467-478.
[5] LI C, YU S, LUO X. Theoretical design and circuit implementation of integer domain chaotic systems[J]. International Journal of Bifurcation and Chaos,2013, 23: 1350170.
[6] LI J, LIU F, GUAN Z H. A new chaotic Hopfield neural network and its synthesis via parameter switchings[J]. Neurocomputing, 2013, 117: 33-39.
[7] SRISUCHINWONG B, AMONCHAILERTRAT N. Realization of a lambert W function for a chaotic circuit[J]. Journal of Circuits, Systems and Computers, 2013,22: 1350075.
[8] LI Q, ZENG H, YANG X S. Simulation of the classical analog phase-locked loop based circuits[J]. Nonlinear Dynamics, 2014, 77: 255-266.
[9] 李清都,唐宋. 三维超混沌映射拓扑马蹄寻找算法及应用[J]. 物理学报,2013, 62: 020510.
[10] ZHOU P, YANG F. Hyperchaos chaos and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points[J]. Nonlinear Dynamics, 2014,76: 473-480.
[11] LI Q D, TANG S, YANG X S. Hyperchaotic set in continuous chaos-hyperchaos transition[J]. Communications in Nonlinear Science and Numerical Simulation, 2014,19: 3718-3734.
[12] GUCKENHEIMER J, OLIVA R A. Chaos in the Hodgkin-Huxley model[J]. SIAM Journal on Applied Dynamical Systems, 2002,1: 105-114.
[13] LI Q, GUO J, YANG X S. Bifurcation and chaos in the simple passive dynamic walking model with upper body[J]. Chaos, 2014, 24: 033114.
[14] YANG X S, LI Q D, CHENG S J. Horseshoe chaos and topological entropy estimate in a simple power system[J]. Applied Mathematics and Computation, 2009,2(11): 467-473.
[15] FAN Q J. Topological horseshoe in nonlinear Bloch system[J]. Chinese Physics B, 2010, 12(19): 120508.
[16] COLLINS S, RUINA A, TEDRAKE R, WISSE M. Efficient bipedal robots based on passive-dynamic walkers[J]. Science, 2005,(7): 1082-1085.
[17] VERDAASDONK B W, KOOPMAN H F J M, van der HEL F C T. Energy efficient walking with central pattern generators: from passive dynamic walking to biologically inspired control[J]. Biological Cybernetics, 2009,1(01): 49-61.
[18] MCGEER T. Passive dynamic walking[J]. International Journal of Robotics Research, 1990,9: 62-82.
[19] LI Q, YANG X S. New walking dynamics in the simplest passive bipedal walking model[J]. Applied Mathematical Modelling, 2012,11(36): 5262-5271.
[20] 李清都,周红伟,杨晓松. 基于异构计算的简单行走模型的吸引区域研究[J]. 物理学报,2012,61: 040503.
[21] LI Q, TANG S, YANG X S. New bifurcations in the simplest passive walking model[J]. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013,4(23): 043110.
[22] FARSHIMI F, NARAGHI M. A passive biped model with multiple routes to chaos[J]. Acta Mechanica Sinica, 2011,27(2): 277-284.
[23] 柳宁, 李俊峰, 王天舒. 双足模型步行中的倍周期步态和混沌步态现象[J]. 物理学报,2009, 58(6): 3772.
[24] GOSWAMI A, THUILOT B, ESPIAU B. A study of the passive gait of a compass-like biped robot symmetry and chaos[J]. International Journal of Robotics Research,1998,17(12): 1282-1301.
[25] GRITLI H, KHRAIEF N, BELGHITH S. Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot[J]. Communications in Nonlinear Science and Numerical Simulation, 2012,17: 4356-4372.
[26] OTT E, GREBOGI C,YORKE J A. Controlling chaos[J]. Physical Review Letters, 1990, 64(11): 1196-1199.